Stolarsky mean

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In mathematics, the Stolarsky mean of two positive real numbers xy is defined as:

& = \lim_{(\xi,\eta)\to(x,y)}
\left({\frac{\xi^p-\eta^p}{p (\xi-\eta)}}\right)^{1/(p-1)} \\[10pt]
& = \begin{cases}
x & \text{if }x=y \\
\left({\frac{x^p-y^p}{p (x-y)}}\right)^{1/(p-1)} & \text{else}

It is derived from the mean value theorem, which states that a secant line, cutting the graph of a differentiable function f at ( x, f(x) ) and ( y, f(y) ), has the same slope as a line tangent to the graph at some point \xi in the interval [x,y].

 \exists \xi\in[x,y]\ f'(\xi) = \frac{f(x)-f(y)}{x-y}

The Stolarsky mean is obtained by

 \xi = f'^{-1}\left(\frac{f(x)-f(y)}{x-y}\right)

when choosing f(x) = x^p.

Special cases[edit]


One can generalize the mean to n + 1 variables by considering the mean value theorem for divided differences for the nth derivative. One obtains

S_p(x_0,\dots,x_n) = {f^{(n)}}^{-1}(n!\cdot f[x_0,\dots,x_n]) for f(x)=x^p.

See also[edit]